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Shelf-life estimation of gun propellant stockpile So Young Sohn Department of Decision Sciences and Engineering Systems, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA (Received 11 November 1994; accepted 20 March 1995)

Gun propellant stability is an important consideration when assessing the safety of gun ammunition. In order to predict deteriorating stability of propellant stockpiles over lifetime, the master sample surveillance program has long been used by U.S. military laboratories. In this paper, we utilize the information obtained from the Navy's master sample surveillance program and employ a random effects linear model to estimate the safe shelf-life of gun propellant stockpiles. Estimation methods are discussed and applied to 5-inch 54-caliber Navy Cool (NACO) propellants. Several recommendations are made to improve the current practice of the surveillance program and subsequent data analyses.

based on the assumption of a linear relationship between the stability and the age, and it does not take into account lot-to-lot variation on the changing patterns of stability. The major concerns of the current practice are (a) if the relationship between the stability and the age of propellant is linear; (b) how to treat potential lot-to-lot variation when estimating the shelf-life; and (c) how reliable it is to use one accelerated condition such as 65.5°C to find unsafe propellants. In this paper, we first introduce the main features of the master sample surveillance program. Some limitations are discussed along with the estimation method currently used for assessing the shelf-life. We then suggest alternative estimation procedures that can accommodate lot-to-lot variation and various accelerated conditions. An actual case is given to illustrate and to compare different estimation procedures. Finally, recommendations are made.

1 INTRODUCTION

Most gun propellant is produced long before its ultimate consumption. 1"2 The stability of a propellant stockpile often deteriorates during storage. 3'4 In order to provide the gun performance required by the U.S. Navy to perform its mission in a safe and economic manner, the Naval Surface Warfare Center, Indian Head divison, has been conducting stability tests of propellant stockpiles. Based on this testing, one of the Navy's goals lies in evaluating the shelf-life of gun propellants for Navy gun ammunition in storage and for the Fleet's stock of loaded ammunition. Since the early 1930's, the military services have monitored the stability of propellant and ammunition stockpiles in what is known as Master Sample surveillance programs. These programs include repeated testing of propellant at an over-stress condition (65.5°C) that accelerates the aging process of propellants. The Navy's program is described in greater detail in Section 2 of this paper. Currently, data from this testing is aggregated over similar propellant stockpiles and is used to fit a simple straight line linear regression model for the stability against the age of propellant. The safe shelf-life is then defined as the age at which 95% of stockpiles are safe. It can be estimated by finding the age at which the lower acceptable level of stability intersects the one-sided 95% prediction limit for the stability. However, the current method of the model fitting is

2 MASTER SAMPLE SURVEILLANCE PROGRAM

In order to evaluate the performance of the gun propellant stockpiles, the master samples have been taken from all propellant lots produced for gun ammunition in the United States. Each master sample consists of 5 lb of an individual propellant lot. The collection of samples is then subjected to the oven test. Details are as follows. 37

38

So Young Sohn

Collected master samples are sent to the laboratory. Initially they remain in real time storage located near the laboratory until a portion of propellant (45 gm) taken from each master sample (5 Ib) is put into the heating chamber at 65.5°C. Its aging is then accelerated until red nitrogen oxide fumes appear. The time it takes until the red nitrogen oxide fumes appear is called the fume time or days-to-fume (DTF). This fume time is used as a measure of the stability of the propellant at the age when it entered the heating chamber. In general, for a given propellant type (same chemical formulation and physical dimensions), older propellant tends to be less stable and associated with a smaller fume time than newer propellant. After the first 45 gm sample is consumed as a result of the accelerated oven test, another 45 gm sample is taken from the remaining master sample stored in the real time storage. It is sent to the heating chamber and the same experiment is repeated. This iterative testing continues until the fume time falls below thirty days. When the fume time falls below thirty days at the accelerated temperature, 65.5°C, the corresponding lot is considered unsafe and the propellant lot is destroyed safely. Typically, it takes a relatively long time (20 to 80 years) to observe such a phenomenon even at 65.5°C and the prediction of the shelf-life is desirable to prevent safety problems and to assist with maintenance, acquisition, and inventory management decisions. The current method used to predict the shelf-life of propellant lots is based on the fixed effects linear regression model for fume time fitted against propellant age collected from a group of similar lots. Normally a group consists of lots provided by the same vendor and manufactured using the same formulation and of the same physical dimensions. Assuming the relationship between the fume time and the age is linear, a straight line is fitted based on the experimental data available at the time when the information regarding shelf-life is necessary (Type I censor). The shelf-life of a group of similar lots is defined as the age when at least 95% of the stockpile is safe. That is, the time period at which a 95% one-sided lower prediction limit for the fume time curve intersects the acceptable lower specification level, 30 days of fume time. This method would be a useful tool to assess the shelf-life of a group of similar lots when the relationship between the fume time and the age is linear and there is no lot-to-lot variation in the group. As for that relationship, Ark et al. 4 indicated that some other patterns based on exponential and polynomial regression might be more suitable. Additionally, in practice, there may be a slight or moderate variation among several lots, which would require the group shelf-life estimation method that takes into account the random variation.

Furthermore, when there are some outlying lots, besides the potential variation, information acquired from a group can be applied to estimate the shelf-life of the individual lot. Since the shelf-life of an individual lot would be estimated using not only its own performance but also that of a group of similar lots, one can obtain the improved accuracy. When the group shelf-life is applied to the maintenance of unstable stockpiles, there is a chance that many individual lots are destroyed, even though they are still safe. In view of such opportunity loss, use of the individual shelf-life may also reduce the cost involved in gun propellant logistics. An additional question raised regarding the current method is about the nature of the accelerated life test used for the master sample surveillance program. In a typical accelerated life test (ALT), 3'5-8 test items (lots) are subjected to two or more severe environmental conditions to accelerate their deterioration. Then there is an effort to relate the deterioration due to accelerated aging with aging that would occur under real life conditions. A major difference of the fume time experiment from typical ALT is that it employs only one level of over-stress (65-5°C). Unlike the typical ALT, performance data (fume time) is not measured at the accelerated aging time but at the actual age at the use condition. Then a pre-set rule (30 days of the fume time at 65.5°C) is adopted for an unacceptable quality for propellant. Applicability of the result obtained from a single stress condition may depend heavily on such a predetermined rule. In view of the risk involved in the single stress experiment, two or three different stress levels could be applied to the fume time experiment by splitting master samples at a cost of reduced sample size for given expense. In this paper, we first present a random effects regression model9-11 which takes into account lot-to-lot variation in fitting the fume time against the propellant age. The random effects model is generalized to facilitate not only the fume time experiment but also a potential use of a typical ALT in which more than one level of over-stress is used.

3 STATISTICAL MODEL

Consider a collection of master samples taken from a group of similar lots manufactured by the same vendor using the same formulation and same physical dimensions. In order to estimate deteriorating patterns of the stability of propellant lots over time, test sample j (j = 1 hi) of lot i (i = 1. . . . , N) is put in the heating chamber at 65.5°C. The corresponding fume time (f,-j) is then recorded against the actual age (xij) of batch j of lot i. In general, it is expected that the fume time decreases as the propellant ages. For many possible presentations of . . . . .

Shelf-life estimation o f gun propellant such a relationship, we employ within-lot model for each lot: For i = 1. . . . , N and j = 1 . . . . , ny, g(fij) =/30;

+/31ih(xq) + 8q

or

the

following

Y0 =/30; +/31~t;j + ~;~ (1)

where g(fo)=yo is the transformed fume time; h(xo) =t;~ is the transformed propellant age; and /30~ and /31~ are intercept and slope coefficients, respectively. Examples of Y0 could be the fume time fq itself or In(f0) while examples of ti~ could be the age x 0 or 1/x o. A selection of appropriate y;~ and t0 will be dictated by the nature of the observed data. A random error ~ is assumed to follow independent normal distribution with mean 0 and variance z 2. That is ~0 ~ Normal(0, z2). In view of potential lot-to-lot variations, we assume /30; and/31,, are random and the following between-lot model specification is assumed: /30; = z;vO + e0;

(2)

/31i = z/~/1 + e l i

where a 1 × k vector z i represents values of k factors that may have influence on varying /30~ and /31i. "r0 and ~,1 are k × 1 vectors of regression coefficients for /30i and/31~, respectively. R a n d o m errors e0; and eli are assumed to follow independent N(0, ~) where £=1_o-20~ tr~J

and they are independent of ~0.

The within-lot and the between-lot models ((1) & (2)) can be re-written using matrix notation, respectively: for i = 1. . . . . N Yi = Td~; + ~i

(3)

situation where more than one experimental condition is employed. Additionally note that the estimation method used currently in the laboratory considers no lot-to-lot variation, i.e., e0~--0 and eli = 0. Therefore eqn (3) becomes Yi = Ti~/+ ~i. (5) An unknown parameter vector, ~/, is estimated based on the combined data set from lot i = 1 to N using the ordinary least square (OLS) method: ~/o : [~/0o, ~/lO], = (T,T)-~(T,y) (6) N

where T and Y are ~ ni × 2 matrices of T / s and Y/s, i=1

respectively. The variance of ~,o, z2(T,T)-~, can be subsequently estimated as var(q °) = (Y - T~/°)'(Y - T~/°) (T'T) -1 .

1

~(x) = g-l[q,0° + ~/l°h(x)l . (8) However, when the model is derived as (3) and (4) to reflect potential lot-to-lot variations, unknown parameters in the within-lot model ( ~ , ~ ) and those in the between-lot model (% I~) should be estimated. We use a two-stage estimation developed by Vonesh & Carter) 2 In the two-stage estimation, the parameters in the within-lot model are estimated separately from those in the between-lot model. A brief summary follows• In the first stage within-lot model, we obtain the OLS estimates of [$;, and

t/1

(7)

Based on this information, ~'i = T~V° or g ~ ( x ) ) = ~l 0 ° + ~/l°h(x) where ~(x) is the fume time of lot i at age x. Therefore, ~(x) can be predicted as

where Yil

39

~ = b; = [bOi, bl,]' = (T'Tg)-~(T'Y,)

(9)

N

X (Y; - Tibi)'(Yi - T;b;)

~ 2 ~ i=1

1

tq

_ Yini

1

tini

I °l

~il

Yi =

[~i ----"

Yo

fll i

T i --

5; =

_

(4)

~ij

fli = Z i ~ "~ I[i

where 0

zi

'Y=

~/1

~i :

The estimated variance matrix of b~ is vfir(b;)-- '~2(T'T;)-I.

(11)

In order to estimate ~/in the between-lot model (4), unobservable [1; is replaced with individual OLS estimates, bi of (9)• Consequently estimation error -q; is added to (4):

_~ ini

Z; =

(10)

Lelif

Note that, in the fume time experiment, all batches are put in the identical condition (65.5°C). Therefore k is 1 and z; is the same for all i's. Nonetheless we consider z; in (2) to accommodate a more general

b; = Zi'I, + ~i + "q; (12) where the mean of a 2 × 1 random error vector "q;=[~10/,'ql;]' is [0,0]' and the variance of "qi, v~r('qi)= z~(T'Ti) -1. The variance matrix of c; + -q; at estimated z2 is V~ = (Y~ + ¢2(T'T~)-x). (13) When X is known, 3' in (12) can be estimated as "Y =

Z;V/-Iz; =

E i=l

Z'V~-lhi

(14)

40

So Young Sohn

and v a r ( q ) = (Yqu=1 Z / V ~ l Z i ) -1. When X is unknown, the maximum likelihood or the restricted maximum likelihood method 9-1~ could be applied for the estimation. However these methods require an iterative procedure which can be computationally burdensome. Additionally, they are sensitive to starting values of estimates, and local convergence problems may occur. Instead we use the corrected method of moment estimator of Z, Z: B ' [ I N -- Z ( Z ' Z ) - I z ' ] B / ( N

- f Z / ( N - k) ~ [1 - z i ( Z ' Z ) - I z f ] ( T [ T i )

-1,

i=1 B ' [ I N --

Z(Z'Z)-~Z']B/(N-

k)

is, the time period at prediction limit for the the acceptable lower fume time. Therefore, the fixed effects model as follows:

30 = g-~[¢/0 ° + ¢/l°h(x °) - t(0"95, i__~n, - 2 )

× VShh(x°)) + ~]

- k)

N

=

turns out to be unsafe. That which a 95% one-sided lower fume time curve intersects specification level, 30 days of the current practice based on (8) estimates the shelf-life (x °)

if ~ < f2

(15)

N

- ~ / ( N - k) ~, [1 - z i ( Z ' Z ) - l z [ l ( T [ T i )

-1,

i=l

if ~ ~ f2 where B is an N × 2 matrix, (bl . . . . . bN)', Z is an N × k matrix, (z~, . . . , Z~v)', and ~ is the smallest root of

(19)

where SZo(h(x°)) = vfir(¢/0° + ¢/l°h(x°)) = [1, h(x°)] vfir(Z/°)[1,h(x°)]'; and t(0.95, N - k ) is the one-sided 95th percentile of the t distribution with N - k degrees of freedom. On the other hand, the group shelf-life (x r) that takes into account lot-to-lot variation (18) can be obtained from the following relationship based on the asymptotic properties of ¢/derived in Ref 12: 30 = g-~[/~0i +/~ 1,h(x r) -/(0.95, N - k)~/SZr(h(xr)) + f2] = g-l[[1, h(xr)]6i -/(0.95, N - k)~/S~(h(xr)) + f2] = g-~[[1, h(xr)]zfil - / ( 0 . 9 5 , N - k)~/S~(h(xr)) + f2]

B'[IN - Z ( Z ' Z ) - a Z ' ] B / ( N - k) - ~](N - k)

(20)

E [1 -- z i ( Z t Z ) - l z [ ] ( T [ T i ) i=l

-1 = O.

When the resultant ~ in (15) replaces Z in (13), ~¢i follows and in turn ¢/=

Z'V;-~Z, "=

~ Z/'v'/-~b~.

(16)

i=1

The estimated variance of the ¢/ is (EL1 Z [ V ? I Z i ) -1. When ¢/ in (16) replaces ~/ in (12), re-estimated as bi = Ziq.

v~r(~)= b i can

be (17)

Finally, by substituting 13i = [li0i, 131i]' in (1) with b i = [~Oi, ~1i]', the predicted fume time of lot i at age x can be estimated as Pt(x) = 301 + ~Lh(x). When yi(x) is re-transformed t o the original scale of fume time f/(x), ~/(X) = g - - l [ ~ 0 i "+"/9l~h(x)]

(18)

where yi(x) = g ( f ( x ) ) .

4 SHELF-LIFE ESTIMATION

The shelf-life of a group of homogeneous lots is defined as the age when at most 5% of the stockpile

where SZ,(h(xO) = v~r(301 +/~ llh(xr)) = [1, h(xr)]zi var(g/)Z/[1, h(xr)] '. The estimated group parameter ¢/ obtained in (16) for a group of similar lots can be used as an input to improve the accuracy of the shelf-life of the individual lot. An individual shelf-life could be predicted based on the individually fitted within-lot model (1) alone. Often, however, the accuracy of the individually fitted model is relatively low compared to that of the fitted model based on a combined data set of a group. This is due to a small number of observations available for each individual lot. In this case an estimator that shrinks individually estimated bi (9) to bi obtained from a group of similar lots (17) can be used to increase the accuracy of the estimation of individual shelf-life. This kind of estimator is also known as a shrinkage estimator. 1°'11'13 A shrinkage estimator (b*) described by Strenio et al. 13 combines information obtained from an individual lot and that from a group of similar lots by taking the weighted mean of bi in (9) and f~; in (17) to obtain b/* = [b0*, bl*]': b* = W,bi + (I - W~)bi

(21)

where W,. = U;-~('tZ(T/Ti)-l) -~ and Ui = [(~2 (T/Ti)-I)-I + ~-~]. Notice that ~ ( T ' T i ) -1 and Y are variance matrices of individual OLS estimates hl and the between-individual [3;, respectively. Therefore the relative weights given to individual bi and G~are proportional to reliabilities of both estimators. Unlike the group shelf-life, the individual shelf-life

Shelf-life estimation of gun propellant of lot i based on the shrinkage estimation, (x*), is then defined as the time period at which the predicted fume time curve intersects the 30 days of fume time: 30 =

g-l[bO* + bl*h(x*)] = g~l[[1, h(x/*)]b/~].

(22)

In summary, we consider two different approaches to estimate the shelf-life of a group of similar propellant lots: (a) the fixed effects model that neglects lot-to-lot variation (19) and (b) the random effects model that takes into account potential lot-to-lot variation (20). Additionally the shrinkage estimation is applied to estimate individual shelf-life (22).

5 DATA ANALYSIS The three methods discussed in the previous section are applied to the master samples taken from each of the twenty eight 5-inch 54-caliber NACO (SPCF) propellant lots in order to assess their group as well as individual shelf-lives. NACO is a generic term used to describe a family of cool-burning, single-base propellants. 2 The name is an acronym for Navy cool propellants. SPCF is an abbreviation of 'Smokeless Powder ethyl-Centralite stabilized Flashless'. The lots Table I. Manufacturing dates of the 5.inch 54-caliber propellant lots Lot

Manufacturing date

11142 11143 11144 11145 11146 11147 11148 11149 11150 11151 11156 11157 11158 11159 11160 11181 11182 11183 11184 11185 11188 11189 11190 11191 11192 11193 11194 11195

10/16 1972 11/061972 11/071972 11/281972 12/071972 04/01 1973 12/27 1972 12/27 1972 12/271972 01/24 1973 01/24 1973 02/15 1973 04/01 1973 04/01 1973 10/18 1973 10/19 1973 04/01 1973 10/24 1973 10/25 1973 07/26 1973 07/29 1973 10/30 1973 10/31 1973 11/01 1973 11/061973 01/14 1974 01/14 1974 01/14 1974

41

are homogeneous in that they were made by Badger Army Ammunition Plant during a similar time period (1972-1974) using the same propellant formulation. As explained earlier, the fume time of each lot was repeatedly observed in an accelerated condition (65.5°C) five to nine times during the last 18 years, which resulted in a total of 221 observations. Table 1 provides the manufacturing dates. A part of the experimental log file is given in Table 2 which includes dates when propellant samples were set in the heating chamber (DATEIN) and associated fume time. Actual propellant age is calculated by subtracting the manufacturing date from the date in which the propellant sample is put in the experimental chamber. Next, in Fig. 1, repeatedly measured fume time is plotted against the propellant age for all 28 lots. In the early age (<1100 days or 3 years), the fume time appears to increase contrary to the general expectation. The increasing pattern of the fume time during the relatively short time period of early age might be explained by the kinetic chemistry of the propellant aging reactions. This may be due to decomposition reactions or other side reactions such as volatilization in the early age. That chemistry is not completely known. However, in general, a negative linear relationship is noticed between the fume time and the age of the propellant except for the early period. It is also noted from Fig. 1 that fume time of individual lots varies at a given age. That is, there are lot-to-lot variations. We proceed with the data analysis as Table 2. Fume time experiment log file Lot ID

Datein Dateout Fume yr/day yr/day time

Lot ID

Datein Dateout Fume yr/day yr/day time

Unit

Julian

Date

Day

Unit

Julian

11142 11142 11142 11142 11143 11143 11143 11143 11143 11144 11144 11144 11144 11144 11145 11145 11145 11145 11146 11146 11146 11146

73/130 77/300 83/228 87/196 73/130 77/230 83/180 87/179 91/183 73/223 77/258 83/187 87/151 90/105 75/151 80/210 85/193 89•028 73/130 77/192 83/152 87/162

75/156 80/322 85/238 89/175 75/124 80/218 85/210 89/175 93/004 75/156 80/240 85/198 89/105 92/181 77/228 83/160 87/146 90/222 75/089 80/142 85/198 89/148

756 1117 741 710 724 1083 761 727 552 663 1077 742 685 806 808 1046 683 559 689 1045 777 717

11142 11142 11142 11142 11143 11143 11143 11143

75/156 80/322 85/238 89/324 75/124 80/218 851210 89/324

77/300 875 83/228 1002 87/196 688 91/172 578 77/230 837 83/180 1058 871179 699 91/183 589

11144 11144 11144 11144

75/156 80/240 85/198 89/105

77/258 833 83/187 1043 87/151 683 90/292 552

11145 77]228 11145 83/160 11145 87/146

801210 1077 85/193 764 89•028 613

11146 11146 11146 11146

77/192 83/152 87/162 90/339

75/089 80/142 851198 89/148

Date

Day

834 1106 694 556

So Young Sohn

42

:,,. t~

.:Z. •

I

I 2000

0

I

I 4000

I

*1 6000

PROPELLANT AGE (UNIT: DAY)

Fig. 1. Fume time vs propellant age: 28 NACO propellant lots.

The four fitted fixed effects models are overlaid in Fig. 2 and the summary statistics are given in Table 3. It is interesting to note that the range of the group shelf-life obtained from each fixed effects model based on (19) is very wide. For instance, model (25) provides the estimated asymptote to be about 412 days of fume time. As a result, the shelf-life is estimated as infinity while model (23) based on the reduced data set provides 23.5 years (8583 days) as the estimated shelf-life. Based on residual analysis, we consider model (23) on the reduced data set as the most appropriate fixed effects model. This model is then used for further analysis. To examine potential lot-to-lot variation, 28 within-lot models are first

follows. First, we use all 221 observations to fit a fixed effects model based on the ordinary least square m e t h o d to see what would have been obtained using the current practice:

f/s =

"t0 + ~/lxjj +

~s.

(23)

Secondly, the same model (23) is fitted based on the reduced data set of 180 observations after eliminating the first three years of observations. Additionally, the reduced data set is used to fit the following two models after transformation of the original variables: In f0 = 3,0 + ~llxis + 8is

(24)

f~s = ~!0 + ~ll/x, s + ~ij.

(25)

"..,.,. '~ %.

"..\ ,.i °%...

•,

•

•

•

,,

..% "..x "... "% .. % "....% , ....% • ....% ,, ....% • ...% ... ,% , .., .% -, ,., "%.

,m

:~':': :'" i O O

I 0

I 20O0

I

I 4OO0

PROPELLANT AGE ( U N I T :

I

I 6OO0

l

I

8000

DAY)

Fig. 2. Four fixed effects models for fume time against propellant age: - - model (23) based on the full data; . . . . model (23) based on the reduced data; ... model (24); - - - model (25).

43

Shelf-life estimation of gun propellant Table 3. Estimated fixed effects models: OLS

Model

~o se(%)

¢/1 se(~l)

Rz

sheff-life (year)

(23) full data (23) reduced data (24)

1045.51040 (23.17296) 1388.45171 (20.61232) 7.37872 (0.05749) 411.73751 (17.41999)

-0.06%2 (0-00575) -0.14061 (0.00462) -0.000179 (0-00001) 1315827 (51305-70612)

0.40

10400 (28.5yr) 8582 (23-5yr) 17400 (47.67yr)

(25)

estimated based on (9), (10) and (11). Results are given in Table 4 and individually fitted models are overlaid in Fig. 3. Residual analysis supports the assumptions made on 60 being an independent Normal variable with a constant variance ¢2= 8107.6177. Next, use of eqns (15) and (16) gives the estimates of the between-lot model parameters as follows: ~/= [1392-3608, -0.141544]'; ~ = [ 19561-176 L-3.705881

-3-705881]. 0.0007743 J'

var(q) = [ 1059.4439 L -0.208463

-0.208463] 0.000046 J"

Table 4. Estimated witll~i~-individual models: OLS i

Lot

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

11142 11143 11144 11145 11146 11147 11148 11149 11150 11151 11156 11157 11158 11159 11160 11181 11182 11183 11184 11185 11188 11189 11190 11191 11192 11193 11194 11195

b0i 1319.81 1281.04 1213.67 1343.29 1311.29 1270.00 1291-15 1405.29 1280.99 1291-43 1303.38 1233.29 1350-43 1265-40 1237"% 1322-33 1264.00 1511.60 1544.80 1722-01 1560-65 1538.82 1551-92 1544-88 1514.30 1522-30 1575.21 1518-36

se(b0i) 110.529 97-168 102.314 109-683 106.360 94.937 98-556 96-476 99-996 98.034 99-112 95.401 98.345 99.528 93.063 95-972 95.116 101-678 101.153 110.138 104.055 101.113 103.036 101.178 100.959 100.395 123.544 99.264

bl,

se(bl~)

-0"12308 -0-11117 -0.09436 -0"13737 -0-12324 -0-12566 -0.11752 -0.16685 -0-12144 -0-12045 -0.13190 -0-10057 -0.14761 -0.12813 -0.11261 -0.13967 -0-11044 -0.16118 -0.17028 -0.19507 -0"16891 -0"16536 -0-17782 -0-16798 -0.15723 -0-16347 -0.18812 -0.16768

0.024970 0.020246 0-021918 0-025672 0-024826 0.021323 0.021094 0.020528 0.021792 0.021150 0-021760 0.020312 0.021951 0-022612 0.021147 0-021785 0.020527 0.023509 0-023421 0.023623 0.023586 0-023346 0.024009 0.023393 0.023345 0-023496 0.037206 0.023437

0.84 0.52

Ho: 60 - N(0, ~) (p-value) (0.004) (0.458) (0-000)

0.79 ~

(0.029)

Consequently, a group shelf-life x r that takes into account lot-to-lot variation can be obtained from the following relationship: 30 = 1392.3608 - 0.141544x r - t(0.95, 27)X/S2(x r) + ¢2. (26) Equation (26) is obtained based on (20) where Zi is replaced with the identity matrix to reflect the constant 65.5°C condition used in the fume time experiment. As displayed in Fig. 4, resulting group shelf-life, x r, turns out to be 8487 days (or 23.25 years) which is about 3 months shorter than what was obtained based on the fixed effects model. Finally, in order to estimate the individual shelf-life, we apply (21) and (22). Obtained W,. is displayed in Table 5 and resulting individual shelf-lives are given in Table 6. Individually estimated shelf-lives (x?) vary from 24 to 30 years. All calculations are done using the P R O C R E G and P R O C IML of a statistical package SAS. 14

6 DISCUSSION

In this paper we discuss the nature of the master sample surveillance program and its use for gun propellant logistics. Main questions raised concerning the current practice are about (a) use of a linear (straight line) model for the fume time and the age of propellant, (b) treatment of potential lot-to-lot fume time variation; and (c) use of a pre-determined rule (30 days of fume time at 65.5°C) to sentence an unsafe propellant lot. An empirical study based on 5-inch 54-caliber N A C O propellant stockpiles indicates the following. As for the relationship, it turns out that the fume time does not necessarily monotonically decrease as the propellant ages. In fact, an increasing pattern is observed in the early period. The blind use of a straight line model fit for the fume time against age should be avoided and the estimated shelf-life based on the straight line model would not be proper. As an

44

So Young Sohn

".... "'"...,... "".,.... I

I

2000

PROP~AGE

I 4000

I

I

(UNIT: DAY)

Fig. 3. F i t t e d w i t h i n - i n d i v i d u a l models: OLS.

alternative, we use several linear models based on the reduced data set that only contains monotonically decreasing patterns. Since the future stability is what concerns reliability practitioners, elimination of unexpected early behavior does not appear to cause a significant problem. However, a consideration of nonlinear models that accommodate the unexpected behavior of the early period is left as one of the future research areas. When individual fume time patterns are compared to each other, in general, some degree of lot-to-lot variation is observed (e.g., see Fig. 1) and use of the random effects model approach is recommended. When the set of individual fume time plotted against

given age (Fig. 1) tends to be pretty much overlapped, the currently used fixed effects model that does not take into account lot-to-lot variation may not provide very different results from the random effects model suggested in this paper. However, note that a condensed collection of individually fitted lines (Fig. 3) does not necessarily indicate insignificant lot-to-lot variations. Although there are significant lot-to-lot variations, it is possible to see such a condensed pattern when the within-lot variance ( r 2) is very large. When apparent outlying patterns are found in some lots one may use a shrinkage estimation to find the individual shelf-lives of those lots. One of the advantages of using the individual shelf-life lies in

I

5oo0

6oo0

7ooo aooo PROPELIANT AGE (UNIT: DAY)

I

9.ooo

zoooo

Fig. 4. G r o u p shelf-life e s t i m a t i o n b a s e d o n t h e r a n d o m effects model: - - p r e d i c t e d f u m e time; . . . a 9 5 % o n e sided l o w e r p r e d i c t i o n limit for a f u m e time.

Shelf-life estimation of gun propellant

Table 6. Estimated individual models: shrinkage method

Table 5. Estimated weight matrix W~ i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Lot 11142 11143 11144 11145 11146 11147 11148 11149 11150 11151 11156 11157 11158 11159 11160 11181 11182 11183 11184 11185 11188 11189 11190 11191 11192 11193 11194 11195

wi(0, 0)

wi(0, 1)

0.85788 0.73118 0.78282 0.92274 0.91269 0.86800 0.78165 0-77113 0.81638 0.79680 0-83013 0.77259 0.86029 0-89380 0.88544 0.88756 0.79728 0-88940 0.89100 0.75384 0-85787 0.88628 0-90452 0.88868 0.88850 0.90740 1.10871 0.91941

1220.58 296-47 678-01 1508.04 1393.51 936.66 587-12 485.76 794.31 651.81 843.00 468-29 974-93 1164-18 980-16 1055.62 587.40 1184-87 1181.19 702.% 1084-40 1157-76 1285.56 1170.64 1156.09 1242.93 2385-18 1275.46

w,(1, 0)

45

wi(1, 1)

-0.000064 0.26570 -0.000026 0.53250 -0.000038 0.44281 -0-000080 0.18512 -0.000078 0.21092 -0-000058 0.36500 -0.000037 0.46153 -0.000035 0.48474 -0.000045 0.41033 -0.000041 0 . ~ 5 -0.000049 0-39648 -0.000035 0.48791 -0-000056 0.00000 -0.000064 0-31136 -0-000063 0.34966 -0.000063 0.33452 -0.000041 0.45724 -0.000072 0.26013 -0.000073 0-26012 -0.000039 0.39958 -0.000064 0-29208 -0.000072 0-26657 -0.000076 0.23464 -0.000072 0-26311 -0.000072 0.26422 -0.000077 0-24115 -0.000159 -0.23094 -0-000080 0.22933

potential economic gain. W h e n the group shelf-life is applied to control the quality of propellant lots, a group of similar lots would be discarded even though the group m a y contain only a few ( 5 % ) substandard individual propellant lots. However, when the individual shelf-life is available, each lot can be treated separately. In view of the potential economic gain due to discarding only those substandard individual lots, use of the shelf-life of the individual propellant lot is expected to be m o r e beneficial than that of the group shelf-life. The use of a pre-determined rule to discard an unsafe propellant lot apparently raises m a n y questions. According to Ref. 4, the lot is considered unsafe when its fume time falls below 30 days at 65.5°C. T h e r e f o r e the group shelf-life was defined as the time period at which the 95% one sided prediction interval for a fume time intersects the 30 days of fume when the experiment is conducted in the 65.5°C oven. H o w e v e r , it was learned through conversation with the reliability practitioner that when the lot is d e e m e d unsafe, occasionally another batch is taken from the same lot and the test is repeated. In this case, it would be m o r e appropriate to consider a lot to be unsafe when its expected fume time, instead of the actual fume time, falls below 30 days. Accordingly, the shelf-life is estimated based on the confidence limit instead of the prediction limit. Apparently, the

i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Lot 11142 11143 11144 11145 11146 11147 11148 11149 11150 11151 11156 11157 11158 11159 11160 11181 11182 11183 11184 11185 11188 11189 11190 11191 11192 11193 11194 11195

b0* 1352.66 1319-97 1284.47 1353-38 1343.87 1301.03 1327.36 1390.04 1317.41 1325.69 1326-63 1288.65 1350-38 1294.50 1284.01 1332.18 1308.29 1475.16 1494.24 1603.23 1507.06 1494.59 1490-05 14%.95 1482.43 1483-02 1484-00 1474.87

b 1"

Life (year)

-0-13201 -0.12249 -0-11394 -0.13684 -0.13138 -0.12862 -0-12667 -0.15426 -0-12823 -0.12803 -0.13338 -0-11592 -0.14138 -0-12921 -0.12174 -0.13651 -0.12204 -0.15527 -0-16011 -0.17588 -0-16035 -0.15837 -0.16219 -0.15951 -0.15449 -0-15684 -0.15993 -0.15765

27-4513 28.8532 30-1652 26.4953 27.3993 27.0737 28.0608 24.1547 27-5062 27.7260 26.6341 29.7472 25.5875 26.8122 28.2200 26.1350 28.6979 25.4992 25.0556 24.5062 25-2365 25.3363 24-6640 25.1966 25.7579 25.3821 24.9081 25-1101

confidence interval is narrower than the prediction interval and the resulting shelf-life would be longer than what was obtained from (19) and (20). Note that the lower confidence limit is obtained by eliminating ¢2 from (19) and (20). For instance, the estimated group shelf-lives of the N A C O propellant stockpile based on the confidence limit concept turn out to be 25-73 and 25-26 years, for the fixed effects and the r a n d o m effects models, respectively. These estimates are longer than their counterparts (23-5 and 23.25 years) obtained from (19) and (20) and would reduce the cost involved in the false alarm. Additionally, in view of the risk involved in the single stress experiment, it is r e c o m m e n d e d that two or three different stress levels be applied to the fume time experiment by splitting a master sample. T h e levels of additional stress could be set after a careful analysis of chemical properties of propellant stockpiles. The master sample (5 lb) will then be split and assigned to each level of stress. For statistically optimal designs for levels of stress and a reduced master sample size for each stress level, see Ref. 7. It is r e c o m m e n d e d that the surveillance tests begin at a relatively later period than the current test to compensate the reduced size of allocated master sample to each stress. W h e n the stability m o d e l is fitted for each level of stress, the results can be extrapolated to the use condition. T h e r e f o r e it is possible to predict the

46

So Young Sohn

stability of a propellant stockpile stored in use condition for a given time. Instead of relying on a pre-set rule to decide the safety of a lot, this information can then be used for the maintenance of propellant stockpiles.

ACKNOWLEDGMENT I wish to thank Mr David Lee from the Naval Surface Weapon Center, Indian Head Division, for providing me with the N A C O data set along with many valuable and constructive comments. I appreciate discussion with Mr Wong Fun Ark and Mr Henry Van Dyke from US A r m y Armament Research Development and Engineering Center, Picatinny Arsenal, which led me to the propellant stability deterioration research.

REFERENCES 1. Salama, J., Long-term stability of navy gun propellants. Technical Report, IHTR 1251, Naval Ordnance Station, Indian Head, MD, 1989. 2. Mayernik, E. & Haukland, A., Quality surveillance evaluation of 5-inch, 54-caliber NACO propellant charges. Technical Report, I-HTR 428, Naval Ordnance Station, Indian Head, MD, 1975. 3. Garman, N. S., Picard, J. P., Polakoski, S. & Murphy, J. M., Prediction of safe life of propellants. Technical Report 4505, Picatinny Arsenal, Dover, NJ, 1973.

4. Ark, W. F., Eccles, C. & Robertson, D., Statistical analysis of ARDEC's fume data base. ARDEC working paper, Picatinny Arsenal, Dover, NJ, 1993. 5. Carey, M. B. & Koenig, R. H., Reliability assessment based on accelerated degradation: a case study. IEEE Trans. Reliab., 40 (1991) 499-506. 6. Merry, G., Type life program history, philosophy, accomplishments, and future. Indian Head Special Report 88-285, Naval Ordnance Station, Indian Head, MD, 1989. 7. Nelson, W., Accelerated Testing, John Wiley & Sons, NY, 1990. 8. Yum, B. J. & Choi, S., Optimal design of accelerated life tests under periodic inspection. Naval Research Logistics, 36 (1989) 779-95. 9. Sohn, S. Y. & Mazumdar, M., A Monte Carlo study of variable selection and inferences in a two stage random coefficient linear regression model with unbalanced data. Comm. Statistics-Theory and Methods, 20 (1991) 3761-91. 10. Searle, S. R., Casella, G. & McCulloch, C. E., Variance Components. John Wiley & Sons, NY, 1990. 11. Bryk, A. S. & Raudenbush, S. W., Hierarchical Linear Models. SAGE Publications, Newbury Park, CA, 1992. 12. Vonesh, E. F. & Carter, R., Efficient inference for random-coefficient growth curve models with unbalanced data. Biometrics, 43 (1987) 617-28. 13. Strenio, J., Weisberg, H. I. & Bryk, A. S., Empirical Bayes estimation of individual growth-curve parameters and their relationship to covariates. Biometrics, 39, (1983) 71-86. 14. SAS, SAS/STAT User's Guide, Version 6, Cary, NC, 1989.

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